Hemodynamic traveling waves in human visual cortex

Functional MRI (fMRI) experiments rely on precise characterization of the blood oxygen level dependent (BOLD) signal. As the spatial resolution of fMRI reaches the sub-millimeter range, the need for quantitative modelling of spatiotemporal properties of this hemodynamic signal has become pressing. Here, we find that a detailed physiologically-based model of spatiotemporal BOLD responses predicts traveling waves with velocities and spatial ranges in empirically observable ranges. Two measurable parameters, related to physiology, characterize these waves: wave velocity and damping rate. To test these predictions, high-resolution fMRI data are acquired from subjects viewing discrete visual stimuli. Predictions and experiment show strong agreement, in particular confirming BOLD waves propagating for at least 5-10 mm across the cortical surface at speeds of 2-12 mm s-1. These observations enable fundamentally new approaches to fMRI analysis, crucial for fMRI data acquired at high spatial resolution.     

Standing waves and traveling waves distinguish two circuits in visual cortex

The visual cortex represents stimuli through the activity of neuronal populations. We measured the evolution of this activity in space and time by imaging voltage-sensitive dyes in cat area V1. Contrast-reversing stimuli elicit responses that oscillate at twice the stimulus frequency, indicating that signals originate mostly in complex cells. These responses stand clear of the noise, whose amplitude decreases as 1/frequency, and yield high-resolution maps of orientation preference and retinotopy. We first show how these maps are combined to yield the responses to focal, oriented stimuli. We then study the evolution of the oscillating activity in space and time. In the orientation domain, it is a standing wave. In the spatial domain, it is a traveling wave propagating at 0.2-0.5 m/s. These different dynamics indicate a fundamental distinction in the circuits underlying selectivity for position and orientation, two key stimulus attributes.     

Existence of traveling waves for integral recursions with nonmonotone growth functions

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c*, and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme (J Reine Angew Math 306:94–121, 1979) for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers. B. Li’s research was partially supported by the National Science Foundation under Grant DMS-616445. M. A. Lewis research was supported by “The Canada Research Chairs program,” and a grant from the Natural Sciences and Engineering Research Council of Canada.     

Human cortical traveling waves: dynamical properties and correlations with responses

The spatiotemporal behavior of human EEG oscillations is investigated. Traveling waves in the alpha and theta ranges are found to be common in both prestimulus and poststimulus EEG activity. The dynamical properties of these waves, including their speeds, directions, and durations, are systematically characterized for the first time, and the results show that there are significant changes of prestimulus spontaneous waves in the presence of an external stimulus. Furthermore, the functional relevance of these waves is examined by studying how they are correlated with reaction times on a single trial basis; prestimulus alpha waves traveling in the frontal-to-occipital direction are found to be most correlated to reaction speeds. These findings suggest that propagating waves of brain oscillations might be involved in mediating long-range interactions between widely distributed parts of human cortex.     

Stimulus motion propels traveling waves in binocular rivalry

State transitions in the nervous system often take shape as traveling waves, whereby one neural state is replaced by another across space in a wave-like manner. In visual perception, transitions between the two mutually exclusive percepts that alternate when the two eyes view conflicting stimuli (binocular rivalry) may also take shape as traveling waves. The properties of these waves point to a neural substrate of binocular rivalry alternations that have the hallmark signs of lower cortical areas. In a series of experiments, we show a potent interaction between traveling waves in binocular rivalry and stimulus motion. The course of the traveling wave is biased in the motion direction of the suppressed stimulus that gains dominance by means of the wave-like transition. Thus, stimulus motion may propel the traveling wave across the stimulus to the extent that the stimulus motion dictates the traveling wave's direction completely. Using a computational model, we show that a speed-dependent asymmetry in lateral inhibitory connections between retinotopically organized and motion-sensitive neurons can explain our results. We argue that such a change in suppressive connections may play a vital role in the resolution of dynamic occlusion situations.     

Landscape mosaic induces traveling waves of insect outbreaks

The effect of landscape mosaic on recurrent traveling waves in spatial population dynamics was studied via simulation modeling across a theoretical landscape with varying levels of connectivity. Phase angle analysis was used to identify locations of wave epicenters on patchy landscapes. Simulations of a tri-trophic model of the larch budmoth (Zeiraphera diniana) with cyclic population dynamics on landscapes with a single focus of high-density habitat produced traveling waves generally radiating outwardly from single and multiple foci and spreading to isolated habitats. We have proposed two hypotheses for this result: (1) immigration subsidies inflate population growth rates in the high connectivity habitat and, thus, reduce the time from valleys to peaks in population cycles; (2) populations in the high connectivity habitat crash from peaks to valleys faster than in an isolated habitat due to over-compensatory density dependence. While population growth rates in the high connectivity habitat benefitted from immigration subsidies, times from population valleys to peaks were greater in high connectivity habitat due to a greater magnitude of fluctuations. Conversely, the mean time of the crash from population peaks to valleys was shorter in high connectivity habitat, supporting the second hypothesis. Results of this study suggest over-compensatory density dependence as an underlying mechanism for recurrent traveling waves originating in high connectivity habitats aggregated around a single focus.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

Spreading speeds and traveling waves in competitive recursion systems

This paper is concerned with the spreading speeds and traveling wave solutions of discrete time recursion systems, which describe the spatial propagation mode of two competitive invaders. We first establish the existence of traveling wave solutions when the wave speed is larger than a given threshold. Furthermore, we prove that the threshold is the spreading speed of one species while the spreading speed of the other species is distinctly slower compared to the case when the interspecific competition disappears. Our results also show that the interspecific competition does affect the spread of both species so that the eventual population densities at the coexistence domain are lower than the case when the competition vanishes.     

Model of traveling waves in a coral nerve network

Coral polyps contract when electrically stimulated and a wave of contraction travels from the site of stimulation at a constant speed. Models of coral nerve networks were optimized to match one of three different experimentally observed behaviors. To search for model parameters that reproduce the experimental observations, we applied genetic algorithms to increasingly more complex models of a coral nerve net. In a first stage of optimization, individual neurons responded with spikes to multiple, but not single pulses of activation. In a second stage, we used these neurons as the starting point for the optimization of a two-dimensional nerve net. This strategy yielded a network with parameters that reproduced the experimentally observed spread of excitation.     

The hypercycle,traveling waves,and Wright's equation

A formal relation between the hypercycle equation and the delay differential equation of E. M. Wright is exhibited using a traveling waves approach. Several unsolved questions in either problem can be related and interpreted, in particular new motivation for the study of Wright's equation is obtained.     

Metabolic synchronization by traveling waves in yeast cell layers

The coordination of cellular behavior is a prerequisite of functionality of tissues and organs. Generally, this coordination occurs by signal transduction, neuronal control, or exchange of messenger molecules. The extent to which metabolic processes are involved in intercellular communication is less understood. Here, we address this question in layers of resting yeast cells and report for the first time the observation of intercellular glycolytic waves. We use a combined experimental and theoretical approach and explain the radial velocity of the waves to arise from the substrate gradient due to local substrate addition. Our results show that metabolic processes introduce an additional level of local intercellular coordination.     

Describing the geographic spread of dengue disease by traveling waves

Dengue is a human disease transmitted by the mosquito Aedes aegypti. For this reason geographical regions infested by this mosquito species are under the risk of dengue outbreaks. In this work, we propose a mathematical model to study the spatial dissemination of dengue using a system of partial differential reaction-diffusion equations. With respect to the human and mosquito populations, we take into account their respective subclasses of infected and uninfected individuals. The dynamics of the mosquito population considers only two subpopulations: the winged form (mature female mosquitoes), and an aquatic population (comprising eggs, larvae and pupae). We disregard the long-distance movement by transportation facilities, for which reason the diffusion is considered restricted only to the winged form. The human population is considered homogeneously distributed in space, in order to describe localized dengue dissemination during a short period of epidemics. The cross-infection is modeled by the law of mass action. A threshold value as a function of the model's parameters is obtained, which determines the rate of dengue dissemination and the risk of dengue outbreaks. Assuming that an area was previously colonized by the mosquitoes, the rate of disease dissemination is determined as a function of the model's parameters. This rate of dissemination of dengue disease is determined by applying the traveling wave solutions to the corresponding system of partial differential equations.     

Spatial spreading of West Nile Virus described by traveling waves

In this work, we propose a spatial model to analyze the West Nile Virus propagation across the USA, from east to west. West Nile Virus is an arthropod-borne flavivirus that appeared for the first time in New York City in the summer of 1999 and then spread prolifically among birds. Mammals, such as humans and horses, do not develop sufficiently high bloodstream titers to play a significant role in the transmission, which is the reason to consider the mosquito-bird cycle. The model aims to study this propagation based on a system of partial differential reaction-diffusion equations taking the mosquito and the avian populations into account. Diffusion and advection movements are allowed for both populations, being greater in the avian than in the mosquito population. The traveling wave solutions of the model are studied to determine the speed of disease dissemination. This wave speed is obtained as a function of the model's parameters, in order to assess the control strategies. The propagation of West Nile Virus from New York City to California state is established as a consequence of the diffusion and advection movements of birds. Mosquito movements do not play an important role in the disease dissemination, while bird advection becomes an important factor for lower mosquito biting rates.     

Learning to cooperate in solving the traveling salesman problem

A cooperative team of agents may perform many tasks better than single agents. The question is how cooperation among self-interested agents should be achieved. It is important that, while we encourage cooperation among agents in a team, we maintain autonomy of individual agents as much as possible, so as to maintain flexibility and generality. This paper presents an approach based on bidding utilizing reinforcement values acquired through reinforcement learning. We tested and analyzed this approach and demonstrated that a team indeed performed better than the best single agent as well as the average of single agents.     

An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation

We examine traveling-wave solutions for a generalized nonlinear-diffusion Fisher equation studied by Hayes [J. Math. Biol. 29, 531–537 (1991)]. The density-dependent diffusion coefficient used is motivated by certain polymer diffusion and population dispersal problems. Approximate solutions are constructed using asymptotic expansions. We find that the solution will have a corner layer (a shock in the derivative) as the diffusion coefficient approaches a step function. The corner layer at z = 0 is matched to an outer solution for z < 0 and a boundary layer for z > 0 to produce a complete solution. We show that this model also admits a new class of nonphysical solutions and obtain conditions that restrict the set of valid traveling-wave solutions. Supported by a National Science Foundation graduate fellowship. This work was performed under National Science Foundation grant DMS-9024963 and Air Force Office of Scientific Research grant AFOSR-F49620-94-1-0044.     

Tonotopically arranged traveling waves in the miniature hearing organ of bushcrickets

Place based frequency discrimination (tonotopy) is a fundamental property of the coiled mammalian cochlea. Sound vibrations mechanically conducted to the hearing organ manifest themselves into slow moving waves that travel along the length of the organ, also referred to as traveling waves. These traveling waves form the basis of the tonotopic frequency representation in the inner ear of mammals. However, so far, due to the secure housing of the inner ear, these waves only could be measured partially over small accessible regions of the inner ear in a living animal. Here, we demonstrate the existence of tonotopically ordered traveling waves covering most of the length of a miniature hearing organ in the leg of bushcrickets in vivo using laser Doppler vibrometery. The organ is only 1 mm long and its geometry allowed us to investigate almost the entire length with a wide range of stimuli (6 to 60 kHz). The tonotopic location of the traveling wave peak was exponentially related to stimulus frequency. The traveling wave propagated along the hearing organ from the distal (high frequency) to the proximal (low frequency) part of the leg, which is opposite to the propagation direction of incoming sound waves. In addition, we observed a non-linear compression of the velocity response to varying sound pressure levels. The waves are based on the delicate micromechanics of cellular structures different to those of mammals. Hence place based frequency discrimination by traveling waves is a physical phenomenon that presumably evolved in mammals and bushcrickets independently.     

Mobile actin clusters and traveling waves in cells recovering from actin depolymerization

At the leading edge of a motile cell, actin polymerizes in close apposition to the plasma membrane. Here we ask how the machinery for force generation at a leading edge is established de novo after the global depolymerization of actin. The depolymerization is accomplished by latrunculin A, and the reorganization of actin upon removal of the drug is visualized in Dictyostelium cells by total internal reflection fluorescence microscopy. The actin filament system is reorganized in three steps. First, F-actin assembles into globular complexes that move along the bottom surface of the cells at velocities up to 10 microm/min. These clusters are transient structures that eventually disassemble, fuse, or divide. In a second step, clusters merge into a contiguous zone at the cell border that spreads and gives rise to actin waves traveling on a planar membrane. Finally, normal cell shape and motility are resumed. These data show that the initiation of actin polymerization is separated in Dictyostelium from front protrusion, and that the coupling of polymerization to protrusion is a later step in the reconstitution of a leading edge.     

Existence and stability of invasion traveling waves for a competition system with random vs.nonlocal dispersals

The purpose of this paper is to investigate asymptotic behaviors of the solutions for a competition system with random vs.nonlocal dispersal.We first prove the existence of invasion traveling waves via using the theory of asymptotic speeds of spread.Then we prove the invasion traveling waves are exponentially stable as perturbation in some exponentially weighted spaces by using the weighted energy and the squeezing technique.     

Competition by allelopathy proceeds in traveling waves: colicin-immune strain aids colicin-sensitive strain

Producing toxic chemicals to suppress both the growth and survivorship of local competitors is called allelopathy; some strains of the bacteria Escherichia coli produce a toxin (named colicin) which may kill colicin-sensitive neighbors while they themselves are immune. In a previous paper, the competitive outcome between colicin-producing and colicin-sensitive strains was shown to differ between a spatially structured and a completely mixed population. In this paper, we analyze the role of a third, "colicin-immune," strain, which does not produce colicin but is immune to it. Without spatial structure, the colicin-immune strain suppresses the colicin-producing strain and enables the colicin-sensitive strain to win. In a spatially structured population, modeled as a reaction-diffusion system, we examine the speed of boundaries between areas dominated by different strains in traveling waves and the events after the collision of two such boundaries. The colicin-immune strain passes through the area dominated by the colicin-sensitive strain and drives the colicin-producing strain to extinction. Subsequently the colicin-sensitive strain occupies the whole population.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

Emergence of coherent traveling waves controlling quadruped gaits in a two-dimensional spinal cord model

 The concerted and self-organizing behavior of spinal cord segments in generating locomotor patterns is modulated by afferent sensory information and controlled by descending pathways from the brainstem, cerebellum, or cortex. The purpose of this study was to define a minimal set of parameters that could control a similar self-organizing behavior in a two-dimensional neural network. When we implemented synaptic depression and active membrane repolarization as two properties of the neurons, the two-dimensional neural network generated traveling waves. Their wavelength and angle of propagation could be independently controlled by two parameters that modulated excitatory premotor neurons and inhibitory commissural neurons. It is further demonstrated that the selection of wave parameters corresponds to the selection of quadruped gaits. Received: 30 July 2001 / Accepted in revised form: 17 April 2002 Correspondence to: A. Kaske (e-mail: alexander.kaske@mtc.ki.se, alexander.kaske@vglab.com)